# Tagged Questions

**20**

votes

**1**answer

454 views

### Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of ...

**6**

votes

**5**answers

369 views

### Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...

**4**

votes

**1**answer

150 views

### To whom is the internal characterization of $Q$-groups due?

A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following:
$G$ is a $Q$-group if ...

**2**

votes

**1**answer

166 views

### Doubly primitive groups with simple socle

The classification of doubly transitive groups with simple socle is
known. A good account of such classification can be found for example
in this paper:
Cameron, Peter J. Finite permutation groups ...

**18**

votes

**2**answers

486 views

### Reference for the triple covering of A_6

I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For $n=6$ or $7$ (and only in ...

**11**

votes

**1**answer

234 views

### Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, g_{k+1}) ...

**6**

votes

**1**answer

437 views

### Automorphism group of a finite group

I would like to ask if there exists an explicit description of $\mathrm{Aut}(G)$, the group of automorphisms of a finite group $G$, in particular, when $G$ is abelian. E.g., if $G = ...

**6**

votes

**1**answer

217 views

### Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...

**2**

votes

**1**answer

129 views

### Subgroup structure of orthogonal groups of small dimension over finite fields

How much is known about the subgroup structure of the orthogonal groups (of dimension n<=7, say) over finite fields? Can anyone point me in the direction of a good reference? I'm aware of a book by ...

**3**

votes

**1**answer

81 views

### On Groups of Maximal Class: Reference

I will be happy if one gives references (oncluding current research) for `classification' (structure) of $p$-groups of maximal class which contain abelian maximal subgroup (i.e. abelian subgroup of ...

**3**

votes

**1**answer

158 views

### Groups with special automorphism group

I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to $H$ is $\sigma$. Is ...

**7**

votes

**2**answers

314 views

### Good effective versions of theorems of Artin and Brauer

The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups.
For example, Artin's theorem is the statement that for every character $\chi$ of ...

**4**

votes

**4**answers

508 views

### A catalog of faithful representations of finite groups?

I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$.
Where ...

**3**

votes

**1**answer

190 views

### Relationship between the number of Sylow subgroups with element orders in finite group

In a finite group what is relationship between the number of Sylow $p$-subgroups with the number of elements of order a multiple of $p$?
Is there any reference for my question?

**4**

votes

**1**answer

230 views

### Finite Unipotent Groups: References

It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}_q)$. (Thanks in advance!!!)
1) The number of ...

**18**

votes

**6**answers

1k views

### Measures of non-abelian-ness

Let $G$ be a finite non-abelian group of $n$ elements.
I would like a measure that intuitively captures the
extent to which $G$ is non-commutative.
One easy measure is a count of the non-commutative ...

**6**

votes

**1**answer

192 views

### Decomposition of $\mathrm{End}(V)$ as $S_n\times S_n$-module

Let $V$ be a finite-dimensional, complex vector space and set $\newcommand{\Gl}{\mathrm{Gl}}G:=\Gl(V)\times\Gl(V)$. Let $E:=\mathrm{End}(V)$ and consider its coordinate ring $\mathbb C[E]$, the space ...

**4**

votes

**0**answers

129 views

### What is known about 2-modular representations of Ree groups of type $F_4$?

A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...

**1**

vote

**1**answer

384 views

### sylow subgroups of GL(n,q)

Dear all
I am interested to know about the r-sylow subgroups of GL(n,q), is there any work about the structure of this subgroups?

**3**

votes

**0**answers

83 views

### On divisors occurring as subgroup sizes

Given a finite group $G$ define $D(G)$ to be the number of divisors $r$ of $|G|$ for which there exists a subgroup of $G$ of order $r$.
Clearly $D(G) \leq d(|G|)$, where $d(n)$ denotes the number of ...

**3**

votes

**0**answers

232 views

### How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?

This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use ...

**3**

votes

**1**answer

276 views

### Character theory of $2$-Frobenius groups.

This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...

**2**

votes

**1**answer

490 views

### Linear algebra of finite abelian groups

If $f: V \to W$ is a surjective homomorphism of vector spaces, and we have fixed a basis for $V$, it is always possible to find a basis for $W$ such that the matrix associated to $\phi$ in the two ...

**11**

votes

**4**answers

695 views

### Realizable Order Sequences for Finite Groups

My post is motivated at least in part by this MO question.
Has there been any work done on realizable order sequences for finite groups? By an "order sequence" I mean a non-decreasing list of the ...

**1**

vote

**0**answers

218 views

### Reference request

I am looking for a reference or proof for the following problem:
Problem: Let $r$ be prime, then $2r$ is a Sylow $p$-number if and only if $2r=1+p^{2^n}$.
Thanks in advance.

**0**

votes

**1**answer

185 views

### What is a “non-splitting covering” of a finite group?

Apologies if this is elementary, but I have never heard the terminology before:
What is a "non-splitting covering" of a finite group?
I encountered the term while reading this paper, in which ...

**14**

votes

**2**answers

526 views

### Maximal number of maximal subgroups

Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...

**10**

votes

**2**answers

320 views

### Convenient reference for subgroups of a finite semidirect product?

Given a finite group $G= H \ltimes N$ (with no particular constraints on $H, N$), it's probably been known for a long time how to describe efficiently the possible subgroups of $G$. A graduate ...

**7**

votes

**4**answers

829 views

### Textbook source for finite group properties deducible from character table?

Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...

**4**

votes

**1**answer

307 views

### Algorithm for Brauer lifting via Brauer tree?

Background: Given a finite group $G$ and a prime $p$ dividing its order, Brauer theory compares the ordinary characters of $G$ with the Brauer characters arising from $p$-modular representations. On ...

**5**

votes

**2**answers

178 views

### Doubly covering an even lattice

I have read that there is a way to construct a group which is a double cover of an even lattice. The very tantalizing thing about this is that if the even lattice is chosen to be the Leech lattice, ...

**8**

votes

**1**answer

285 views

### Reference for restriction of a simple module over a splitting field to a smaller field?

This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group ...

**5**

votes

**1**answer

520 views

### What is the name for a finite-group representation that is the sum of all the irreducibles (once)?

I vaguely remember seeing a paper studying the concept of a totally multiplicity-one representation of a finite group, which concept, I recall, had a particular name, which I forget. What is this ...

**13**

votes

**3**answers

597 views

### Reference for representation theory of SL_2(Z/n)

There are many references for the representation theory (say over $\mathbf C$) of $SL_2(\mathbf{F}_q)$ and $GL_2(\mathbf{F}_q)$, for instance lecture 5 in Fulton--Harris "Representation theory" and ...

**5**

votes

**2**answers

417 views

### abelian centralizers in almost simple groups

Hallo!
I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.
Let $S$ be a non-abelian finite ...

**41**

votes

**1**answer

2k views

### Which small finite simple groups are not yet known to be Galois groups over Q?

The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...

**8**

votes

**1**answer

342 views

### Radical of $F_p[SL(2,p)]$

Let $G=SL(2,p)$. Does anyone know what is the radical of the group algebra $F_p[G]$?
Does there exists any book/paper where it is calculated?
By radical here I mean maximal ideal I of $F_p[G]$ such ...

**7**

votes

**0**answers

299 views

### The maximal order of an element in orthogonal groups over finite fields of characteristic 2

Let $q$ be a power of $2$ and let $(V,Q)$ be a
quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type ...

**4**

votes

**1**answer

249 views

### Unpublished Higman manuscript

In his paper 'Natural constructions of the Mathieu groups,' Curtis references an unpublished manuscript of G. Higman with a "significant constuction which makes use of the outer automorphism of $S_6$ ...

**19**

votes

**0**answers

497 views

### Is there a Mathieu groupoid M_31?

I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...

**9**

votes

**2**answers

968 views

### Orders of automorphism groups of p-groups

There is a theorem that says that if $p$ is a prime and $G$ is a $p$-group with $|G| = p^{n}$, $|Aut(G)|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$.
This theorem is sharp, since $\Pi_{k=0}^{n-1} ...

**22**

votes

**4**answers

2k views

### Character free proof that Frobenius kernel is a normal subgroup?

The question is in the title, but here is some background/reminders:
A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash ...

**5**

votes

**1**answer

247 views

### Group not leaving subset invariant

Let $Y,X$ be two sets of size n,m. Let $Y\subset X$.
What is the maximal group(in size) $G< Sym(X)$ such that gY=Y imply that $g=1$?
Here I mean that the only permutation which permutes elements of ...

**10**

votes

**4**answers

721 views

### Efficient presentations for finite groups

A finitely presented group which has more generators than relations has an infinite abelianization and so is an infinite group. Therefore, for a finite group, all presentations must have at least as ...

**2**

votes

**1**answer

340 views

### Finite completely reducible groups-reference request

A group G is called completely reducible if it is a direct product of simple groups. It is known that a Krull-Schmidt Remak type unicity for the decomposition in the direct product of simple groups ...

**2**

votes

**1**answer

454 views

### Unique factorization of finite groups under direct sum?

I am told that finite groups have unique factorization under direct product. That is, call a nontrivial group "indivisible" if it is not isomorphic to a direct product of nontrivial groups. Then ...

**1**

vote

**0**answers

195 views

### Pierpont primes

A Pierpont prime is a prime $p$ that can be written as $$p=2^u 3^v + 1.$$
What is known about Pierpont primes? I'm not a number theorist, and the best I can find is
...

**6**

votes

**1**answer

578 views

### Character table for the affine group of Z/p^nZ

Initial caveat: the following question could probably be answered by Google, MathSciNet or my library, if I could find the right search terms or book... but I've not had any luck today, so I hope ...

**13**

votes

**0**answers

491 views

### How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?

This is a follow-up to a recent mathoverflow question
34387
about computing the orders of finite unitary groups and the comments made there.
Between 1955 (Chevalley's Tohoku paper) and 1968 ...

**3**

votes

**4**answers

715 views

### List of centers of finite groups

What I'm really looking for is a good set of examples for a semi-direct product of two (coprime) cyclic groups, with a non-trivial (not $1$ and not everything) center. What resource would you ...